Simulation of Room Impulse Response by using Hybrid Method of Wave and Geometrical Acoustics

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1 ISSN Simulation of Room Impulse Response b using Hbrid Method of Wave and Geometrical Acoustics Wang Hong-wei,Huang Kun-peng Institute State Ke Laborator of Subtropical Building Science, South China Univ. of Tech., Guangzhou , China *corresponding author: wanghw@scut.edu.cn (Received 30 November 2009; accepted 7 December 2009) More and more attention has been paid to the auralization during the past few ears. In this paper, the auralization technique was reviewed at first. Then, to realize the hbrid broad-spectrum simulation in the auditorium including the low frequenc, an improved hbrid algorithm, which combines geometric acoustics and the wave acoustics method, has been proposed. The middle and the high frequenc of the impulse response were simulated b using Odeon software, while the low frequenc of the impulse response was simulated through the FDTD method. The method can deal with trul lower frequenc bands simultaneousl b using the low frequenc. Furthermore, the phase of the impulse was reconstructed through the Hilbert transform. The results validated the applicabilit of the improved method and the have also shown that FDTD can improve auralization, although it not alwas create more virtual sound field. Ke words: Subjective preference, Auralizationg;impulse response;convolution;subjective experiment;phase reconstruction;fdtd 1. INTRODUCTION More and more attention has been paid to the auralization during the past few ears. The foundation of the auralization is Computer simulation of the room acoustics. Computer modeling of room acoustics was proposed during the 1960s b Schroeder et al. [1] and was used in practice b Krokstad et al. [2] and Schroeder [3]. The computer modeling of room acoustics has been traditionall limited to the ra tracing method and image source method. These are high-frequenc techniques based on geometrical acoustic theor, and the developed rapidl in the past 40 ears. However, man wave phenomenons are neglected in these simulations. Till now, it is well known that such a situation is not valid in low frequenc. During the recent decade, low frequenc simulation has been investigated b man authors and its importance and application in the modeling have been pointed out [4 8]. The investigations carried out b Kuttruff [9], Hodgson[5], Dalenback et al. [6] and Lam [7] have shown that the low frequenc impulse response not onl affect the accurac of the calculation of acoustical parameters such as sound pressure levels and reverberation time, but also have influence on the qualit of auralization. The ra-tracing and the sound image method is onl appropriate in the middle-frequenc range. Direct time-domain approaches seem can be used to solve the low frequenc of the impulse response more appropriatel. In this paper, in order to investigate the low frequenc phenomenon, the true wave nature of sound must be considered. FDTD method is used for solving the low-frequenc range problem. FDTD method means solving approximatel within the finite difference time-domain. FDTD simulation is not popular for solving problems in linear acoustics. In contrast, it is one of the ver few techniques used to solve nonlinear wave equations that govern computational fluid mechanics. [10]FDTD is a digital computation technique based on wave equation. This method meets most of the demands required for a good brute force numerical solution of the problem: All calculations are done directl in time domain, acoustical equations are locall resulting in an explicit formulation, and the numerical formulation is itself conservative. A direct calculation in time domain has the advantage of speed, but poses some problems for describing frequenc-dependent material characteristics, such as boundar conditions. Until now there is not a well developed FDTD for the room acoustics. This paper is organized as follows. First the simulation basics are reviewed, especiall within the low frequenc range. Then J. Temporal Des. Arch. Environ. 9(1), December, 2009 Wang and Huang 9

2 the geometric acoustics method was used to calculate the impulse response at the middle-frequenc. Further more; FDTD was used to calculate the impulse response at the low-frequenc. Finall, the two different methods were snthesized to improve the accurac of the FDTD simulation within low-frequenc. It will be shown how the simulation method can be used as an amendment to ra tracing for low frequencies order in the ODEON software is set to 6, so after order 6 the ras reflection directions are chosen at random distribution following the Lambert's law. The impulse response sampling is set at 0.1ms. 2. Full Band Impulse response simulation 2.1 Middle-frequenc simulation There is a number of different modeling techniques currentl used to predict the acoustics of the auditoria. Man prediction methods based on geometrical acoustics allow for some inclusion of the effects of diffuse reflection. The most popular one include ra tracing, the image method, and various forms of beam tracing. Ra tracing [11] creates a dense spread of ras, which are subsequentl reflected around a room and tested for intersection with a spherical detector. The energ attenuation of the ras and distances traveled are used to construct the echogram. Although relativel simple to implement, the algorithm has inherent sstematic errors whereb spurious reflections can be created due to using a non point detector whilst other valid reflections are missed as the ras diverge [12]. Hence image method can provide a statistical evaluation of a sound s likel incidence at a spherical representation of the receiver; the image method can provide the exact information at a point receiver. This method is of questionable utilit for detailed echogram prediction and for auralization even a substantiall higher number of ras are used. The reason for this is that the geometrical acoustic can not accuratel solve the lower frequenc impulse response. In realit, spatial and temporal of low frequenc can onl be solved in different was. In this paper, we have used the software to simulate the middle frequenc of the impulse response. The FDTD method was onl used as supplementar simulation of the low frequenc. There three principal parameters which have to be chosen when carring out a calculation in ODEON4.0, and which can have a significant effect on the results. Although a lot more experience is needed before definitive rules can be laid down, it is possible to give some indication of how these should be set for best results. Figure 1 shows the calculation model. In the figure, ras are followed up to the sixth reflection order. The transition Figure 1. The model of simulation in ODEON.(The red is the source,the blue is the receiver) The reflect-gram displas the arrival of earl reflections to a receiver. When the earl reflections are calculated from detected image sources, it follows that each single reflection can be separated independentl. The single reflection in this paper was set to 6 according to the transition order. In addition to arrival time and energ of the reflection, it is also possible to get information about the direction and which surfaces are involved in the reflection path. Figure 2 shows the simulation result of the impulse response on the geometrical acoustics. From the fig.2, the middle of the impulse response of the listen room can be simulated. The amplitude of the impulse response has been normalized. The impulse responses show the earl reflection in detail. The frequenc responses can be calculated through the FFT transform. Figure 3 shows the frequenc response of the full band of a listen room. Figure 2. The impulse response of the simulation in middle frequenc. Natural Science Foundation of Guangdong Province; ina: J. Temporal Des. Arch. Environ. 9(1), December, 2009 Wang and Huang 10

3 Locall reacting surface (LRS) is supposed on the boundar, where the normal component of the particle velocit at the surface of the wall depends on the sound pressure in front of the wall element and not on pressure in front of neighboring element. On the absorbing boundar conditions, there is a relationship between the normal component of particle velocit on the boundar ( u n ) and its nearest sound pressure ( p ) p u n boundar = (5) Z n Figure 3. The frequenc response of the simulation in Odeon. 3. FDTD Simulation of Room s Low-Frequenc Impulse 3.1 The FDTD simulation Theor The Cartesian staggered grids are used in the simulation here, i.e., in ever grid position the acoustical pressure is half step awa from its nearest particle velocit components. We should know that the time steps between pressure and particle velocit components in each grid position are also staggered, as we can see in the equations below. For the sake of simplicit, we suppose that the spatial discrete steps of three different directions are the same in distance δ x = δ = δ z = δ h, hence the FDTD approximations to the equations of linear acoustics in air become [13] [ l + 0.5] [ l 0.5] ux ( i + 0.5, j, k) = ux ( i + 0.5, j, k) δ t [ l] [ l] p ( i + 1, j, k) p ( i, j, k) u ( i, j + 0.5, k ) = u ( i, j + 0.5, k) l l + [ 0.5] [ 0.5] δt [ l] [ l] p ( i, j + 1, k) p ( i, j, k) [ l + 0.5] [ l 0.5] uz ( i, j, k + 0.5) = uz ( i, j, k + 0.5) δt [ l] [ l] p ( i, j, k + 1) p ( i, j, k) 2 [ l+ 1 ] [ l] ρ0c δ t [ l+ 0.5] [ l+ 0.5] p ( i, j, k) = p ( i, j, k) { ux ( i 0.5, j, k) ux ( i 0.5, j, k) δ + [ l+ 0.5 ] [ l u (, 0.5, ) ] i j k u ( i, j 0.5, k ) + [ l+ 0.5 ] [ l u (,, 0.5 ) ] z i j k uz ( i, j, k 0.5) + } where positions. The upper index l is also integer and stands for the discrete time. For the simulation here, we suppose the local air ρ 3 densit 0 equals 1.21 kg / m and the local sound speed equals 344 m / s. 3.2 The FDTD Boundar Conditions (1) (2) (3) (4) i, j, k are integers and stand for the spatial where Z n is the normal acoustic impedance on the boundar, and[14] ( 1 1 ) / ( 1 1 ) Zn = ρ 0 c + α α where α is the normal sound absorption coefficient on the boundar. 3.3 The Stimulus of The FDTD For the room impulse response (RIR) can reveal most of the acoustic characteristics of a room, an impulse source is given as an initial condition. The frequentl used source is Gaussian impulse source, which can be expressed as ( ) 2 p( t) = A( t t )exp t t / T (7) Where t 0 is the time when the impulse wave node exists, and the magnitude of A determines the impulse s amplitude. The Gaussian width T 0 is used to determine the frequenc range excited b the Gaussian impulse, and the width of T 0 is determined b the spatial discrete step δ h, and the spatial steps are limited b computer s capabilit and constrained b a stabilit condition: δt δ h /( 3 c). Generall the spatial δh step is chosen among the range of 1/10 to 1/20 of the relevant wavelength. 3.4 Model of the Listen Room in the FDTD Considering the irregularit of wall and the interested frequenc range, the FDTD mesh cell size is x = = z = 3cm, and the time step is determined b the 3D stabilit condition: t = h /( 3 c),where h is spatial step and c is sound speed. FDTD simulation capabilit depends on fineness of mesh grids. For the mesh grids mentioned above, such model can simulated sound propagation of frequenc up to 1kHz, considering spatial step of mesh should be less than one tenth of the smallest wavelength. First a model of listen rooms is established. The listen room has the volume of 7.62m b4.25m b 3.95m. The source model of equation (7) is set at the position of S (2.2, 6.6, 1.2), that is 2.2m b length,6.6m b width and 1.2m b height. In order to investigate different positions acoustics, (6) J. Temporal Des. Arch. Environ. 9(1), December, 2009 Wang and Huang 11

4 we also place some receivers in the listen room, and the are P1(2.2, 1.0, 1.2), P2(2.2, 1.6, 1.2), P3(2.2, 2.2, 1.2), P4(1.6, 1.0, 1.2), P5(1.0, 1.0, 1.2). 4. The Impules Response Snthesis Through The Hilbert Transform The impulse response can be transformed from the time domain into the frequenc domain. The impulse responses with different frequenc scope have been calculated in different wa. In this paper, different frequenc response scope has been snthesized. Some facts about the Hilbert transform are stated here without proof [15]. Further the Hilbert transform is linear. Consequentl, for an function for which a Fourier transform exists. The impulse response with the minimum phase can be reconstructed from the snthesized frequenc response. We can Figure 2 shows a middle frequenc impulse response x (t), and its FFT transform, X (t) which was simulated through the traditional method. Figure 4 shows a low frequenc impulse response x (t), and its FFT transform, X (t),which was simulated through FDTD method. We used the lower frequenc response that simulated through the FDTD method as the full band. The lower cut of the snthesized frequenc is set at the 315 Hz, because the FDTD method simulation s limitation. The middle frequenc response was used the traditional method. The lower frequenc energ of the traditional method was filtered out. Figure 5 shows the snthesis result of the full band frequenc impulse response. The amplitude of the frequenc has been constructed after the snthesis. The amplitude space and the phase space makes up the Hilbert space. The minimum phase space can be reconstructed through the Hilbert transform. The Hilbert transform was computed b using the same amplitude From the fig 5,we can see the amplitude of the snthesized frequenc. Figure 5. The ssthesised frequenc response with full band. 5. CONCLUSIONS The subject of this paper is simulation method in the low frequenc. The FDTD method was used to simulate the impulse response in the low frequenc. The Hilbert transform has been used in the phase reconstruction. A real listen room s response has been use to test the reconstruction s fidelit. The subjective evaluation has been carried to verif the sntheses. In our investigation above, we find that the low frequenc impulse response simulated b the FDTD has great space information than that simulated b the traditional method. The two low frequenc impulse response from different method have distinctl difference. The further stud on the phase reconstruction should be done to testif the low frequenc impulse response simulation. And more research work on experiments on the accurac and efficienc for the simulation of full frequenc bands will be done in the future. Acknowledgment The project was supported b Doctoral Fund of Ministr of Education of China. The project was also supported b Natural Science Foundation of Guangdong Province; China. Thanks must also be given to the reviewers for their help of the improvement of this paper. Figure 4. The simulated frequenc response used FDTD method REFERENCES [1] Borish J. Extension of the image model to arbitrar polhedra. J Acoust Am 1984(75): [2] Krokstad A, Strφm S, Sφsdal S. Fifteen ears experience with computerized ra tracing. Appl.[5] Hodgson MR. Evidence of diffuse surface reflection in rooms. J Acoust Soc Am 1991;89: [3] Schroder,M.R,New Method of Measuring Reverbration times. J Acoust Soc Am 196,37: [4] Kuttruff H. Room Acoustics [M]: The Second Edition. London: Applied Science Publishers Ltd. 1979: J. Temporal Des. Arch. Environ. 9(1), December, 2009 Wang and Huang 12

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